Subject: [mg131626] Re: Problem with change of variables in an integral

From: Youngjoo Chung <ychung12 at gmail.com>

Date: Fri, 13 Sep 2013 00:35:35 -0400 (EDT)

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Dear Alexei,
I believe the references mentioned in Alex's reply below explains the evaluation procedure well.
In your earlier message, with
f[z_] := 1/(1 + z^3);
Integrate[...] is evaluated to (2 \[Pi])/(3 Sqrt[3]) before being passed to Map. This explains the difference that you saw.
If f[z] is not known, Integrate cannot evaluate and is passed to Map as is as shown in your message. However, the following code
Clear[f, \[Phi]];
Map[ReplaceAll[#, {z -> r*Exp[I \[Phi]], \[Phi] -> 2 \[Pi]/3}] &,
Integrate[f[z], {z, 0, \[Infinity]}]]
% /. f[z_] -> z
issues an error message
Integrate::ilim: Invalid integration variable or limit(s) in {E^(I \[Phi]) r,0,\[Infinity]}. >>
and even though f[z_] = z is integrable, integration will not be done since E^(I \[Phi]) is not a valid integration variable.
Sincerely,
Youngjoo